Worm and Gear Constraint

Description

The Worm and Gear Constraint block represents a kinematic constraint
between worm and gear bodies held at a right angle. The base frame port identifies the
connection frame on the worm and the follower frame port identifies the connection frame
on the gear. The rotation axes coincide with the connection frame
z-axes. The worm and gear rotate at a fixed velocity ratio determined
by the gear pitch radii or tooth-thread ratio.

The worm thread direction can follow either right-hand or left-hand conventions. The
convention used determines the relative directions of the worm and gear rotational
velocities. A right-hand convention causes the worm and gear to rotate in the same
direction about the respective z-axes. A left-hand convention causes the worm and gear
to rotate in opposite directions instead.

The block represents only the kinematic constraint characteristic to a worm-and-gear
system. Gear inertia and geometry are solid properties that you must specify using
Solid blocks. The gear constraint model is
ideal. Backlash and gear losses due to Coulomb and viscous friction between teeth are
ignored. You can, however, model viscous friction at joints by specifying damping
coefficients in the joint blocks.

Gear Geometry

The rack-and-pinion constraint is parameterized in terms of the dimensions of the
worm and gear pitch circles. The pitch circles are imaginary circles concentric with
the worm and gear bodies and tangent to the thread contact point. The pitch radii,
labeled RB and
RF in the figure, are the radii
that the worm and gear would have if they were reduced to friction cylinders in
mutual contact.

Gear Assembly

Gear constraints occur in closed kinematic loops. The figure shows the closed-loop
topology of a simple worm-and-gear model. Joint blocks connect the worm and gear
bodies to a common fixture or carrier, defining the maximum degrees of freedom
between them. A Worm and Gear Constraint block connects the worm and
gear bodies, eliminating one degree of freedom and effectively coupling the worm and
gear motions.

Assembly Requirements

The block imposes special restrictions on the relative positions and orientations
of the gear connection frames. The restrictions ensure that the gears assemble only
at distances and angles suitable for meshing. The block enforces the restrictions
during model assembly, when it first attempts to place the gears in mesh, but relies
on the remainder of the model to keep the gears in mesh during simulation.

Position Restrictions

The distance between the base and follower frame
z-axes, denoted dB-F in the
figure, must be equal to the distance between the gear centers.

The translational offset between the base and follower frame origins
along the follower frame z-axis, denoted
ΔZF in the figure, must be zero.

Orientation Restrictions

The z-axes of the base and follower frames must be
perpendicular to each other. The z-axes are shown in
blue in the figure.

The cross product of the follower frame z-axis with
the base frame z-axis must be a vector aimed from the
follower frame origin to the base frame z-axis. The
z-axes and their cross-product vector are shown
in the figure. The cross product is defined as z^F×z^B.

B — Base frameframe

F — Follower frameframe

Parameters

Winding direction of the worm thread relative to the base frame z-axis. As
viewed from the base frame origin, a right-hand thread is one that wraps
around the base frame z-axis in a counterclockwise direction. A left-hand
thread is one that wraps in a clockwise direction. This parameter determines
the relative directions of motion of the worm and gear bodies.

Worm Lead Angle — Angle between the worm thread and rotation plane10deg (default) | positive scalar between 0 and 180
in units of angle

Angle between the tangent to the worm thread and the plane perpendicular
to the base frame z-axis. The lead angle impacts the gear rotation
corresponding to a full worm revolution.